There has been a great deal of theoretical work on sophisticated charged current quasi-elastic (CCQE) neutrino interaction models in recent years, prompted by a number of experimental results that measured unexpectedly large CCQE cross sections on nuclear targets. As the dominant interaction mode at T2K energies, and the signal process in oscillation analyses, it is important for the T2K experiment to include realistic CCQE cross section uncertainties in T2K analyses. To this end, T2K's Neutrino Interaction Working Group has implemented a number of recent models in NEUT, T2K's primary neutrino interaction event generator. In this paper, we give an overview of the models implemented and present fits to published νμ and νμ CCQE cross section measurements from the MiniBooNE and MINERνA experiments. The results of the fits are used to select a default cross section model for future T2K analyses and to constrain the cross section uncertainties of the model. We find strong tension between datasets for all models investigated. Among the evaluated models, the combination of a modified relativistic Fermi gas with multinucleon CCQE-like interactions gives the most consistent description of the available data.